Existence

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1. #1

   (ii) Starting from a different representative :

   $${\mathcal{E}}^{\prime }(h)=\sum W_t(i,j)d_G(g^{\prime h}(i,j),e{)}^2=\sum W_t(i,j)d_G(g^{h_0\cdot h}(i,j),e{)}^2=\mathcal{E}(h_0\cdot h).$$
2. #2

   Proof. Parametrise and . The flattening energy is:

   $$\mathcal{E}(\phi )=\sum _{\begin{array}{c}i,j\in F^{\circ }\\ W_t(i,j)>0\end{array}}{W}_t(i,j)(1-\cos ({\alpha }_{ij}+{\phi }_i-{\phi }_j)).$$
3. #3

   Proof. Parametrise and . The flattening energy is: Step 1 (Exact convexity on the torus). Define . The energy is:

   $$\mathcal{E}(\phi )=\sum _{(i,j)}{W}_t(i,j)(1-\cos {\theta }_{ij}).$$
4. #4

   Step 2 (Characterisation of critical points). At any critical point, for all :

   $$\sum _{j\sim i}{W}_t(i,j)\sin ({\alpha }_{ij}+{\phi }_i-{\phi }_j)=0\forall i\in F^{\circ }.$$
5. #5

   Step 4 (Hessian analysis and uniqueness). With all established, the Hessian of at has entries:

   $$H_{ii}=\sum _{j\sim i}{W}_t(i,j)\cos {\theta }_{ij}^{\ast },H_{ij}=-W_t(i,j)\cos {\theta }_{ij}^{\ast }(i\ne j).$$
6. #6

   For : The optimal gauge solves a graph Laplacian system (Theorem 7.11). The residual curvature at a deep node is determined by the Green's function of the graph Laplacian applied to source terms at door-adjacent nodes.

   $${\rho }_{F^{\circ }}(i)\le C{e}^{-\beta d_{\mathrm{graph}}(i,\Sigma )}\cdot \sum _{p\in \Sigma }{e}_p$$
7. #7

   Quantitatively:

   $$\sum _{i\in F_{\mathrm{deep}}}\rho (i)\le \alpha \cdot {\mathcal{E}}_{F^{\circ }}(h^{\ast })(\alpha <1),$$
8. #8

   Step 4 (Lojasiewicz convergence). By the Lojasiewicz gradient inequality (Fact B.8) applied to the real-analytic function on the compact manifold : there exist and such that near any critical value :

   $$\Vert \mathrm{grad}\mathcal{E}(h)\Vert \ge C|\mathcal{E}(h)-c{|}^{\alpha }.$$